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Lower Ricci curvature and nonexistence of manifold structure

Erik Hupp, Aaron Naber and Kai-Hsiang Wang

Geometry & Topology 29 (2025) 443–477
Abstract

We build limit spaces (Mjn,gj) (Xk,d) of manifolds Mj with uniform lower bounds on Ricci curvature such that Xk is nowhere a topological manifold, and in fact every open set U X has infinitely generated homology.

More completely, it is known that any such Xk must be k-rectifiable for some unique dim X := k n = dim Mj. It is also known that if k = n, then Xn is a topological manifold on an open dense subset, and it has been an open question as to whether this holds for k < n. Consider now any smooth complete 4-manifold (X4,h) with Ric > λ and λ . Then for each 𝜖 > 0 we construct a complete 4-rectifiable metric space (X𝜖4,d𝜖) with dGH(X𝜖4,X4) < 𝜖 such that the following hold. First, X𝜖4 is a limit space (Mj6,gj) X𝜖4, where Mj6 are smooth manifolds satisfying the same lower Ricci bound Ric j > λ. Additionally, X𝜖4 has no open subset which is topologically a manifold. Indeed, for any open U X𝜖4 we have that the second homology H2(U) is infinitely generated. Topologically, X𝜖4 is the connect sum of X4 with an infinite number of densely spaced copies of P2.

In this way we see that every 4-manifold X4 may be approximated arbitrarily closely by 4-dimensional limit spaces X𝜖4 which are nowhere manifolds. We will see that there is a sense, as yet imprecise, in which generically one should expect manifold structures to not exist on spaces with higher-dimensional Ricci curvature lower bounds.

Keywords
Ricci lower bounds, collapsing, non-manifold structure
Mathematical Subject Classification
Primary: 53B20, 53C21
Secondary: 51F30, 53C23, 58C07
References
Publication
Received: 3 September 2023
Revised: 2 February 2024
Accepted: 9 March 2024
Published: 22 January 2025
Proposed: Tobias H Colding
Seconded: Dmitri Burago, Bruce Kleiner
Authors
Erik Hupp
Department of Mathematics
Northwestern University
Evanston, IL
United States
Aaron Naber
Institute for Advanced Study
Princeton, NJ
United States
Kai-Hsiang Wang
Department of Mathematics
Northwestern University
Evanston, IL
United States

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